Cluster-critical class characterization of clustered chromatic number

Let $\mathcal{M}$ be a minor-closed class of graphs, and let $k\geqslant 0$ be an integer. A graph class is $k$-cluster critical if it is obtained as a wedge-product of $v$ copies of $\mathcal{I}$ and $p$ copies of $\mathcal{P}$, in some order, with $v+2p=k+1$. Cluster-critical conjecture.

$$\chi_{\star}(\mathcal{M})\geqslant k+1$$

if and only if $\mathcal{G}\not\subseteq\mathcal{M}$ for some $k$-cluster critical class $\mathcal{G}$. This reformulates the preceding obstruction conjecture in terms of wedge-products and cluster-critical classes. The source presents it as a conjectural characterization; no general proof or refutation is given.